Synchronization of complex dynamical networks with time delays

In the present paper, two kinds of dynamical complex networks are considered. The first is that elements of every node have different time delays but all nodes in such networks have the same time-delay vector. The second is that different nodes have different time-delay vectors, and the elements of each node also have different time delays. Corresponding synchronization theorems are established. Numerical examples show the efficiency of the derived theorems.     

Eigenvalue ratios of vibrating strings

Summary New results for the eigenvalue ratios of vibrating strings are presented partially in connection with previous results concerning Schr?dinger operators.     

Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics

A nonlinear stochastic dynamical model on a typical HAB algae diatom and dianoflagellate densities was created and presented in this paper. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. The singular boundary theory of diffusion process and the invariant measure theory were applied in analyzing the bifurcation of stability and the Hopf bifurcation of the stochastic system. The critical value of the stochastic Hopf bifurcation parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.     

Existence of energy minimizing vortices attached to a flat-top seamount

The existence of an energy minimizer relative to a class of rearrangements of a given function is proved. The minimizers are stationary and stable solutions of the two-dimensional barotropic vorticity equation, governing the evolution of geophysical flow over a surface of variable height. The theorem proved implies the existence of a family of stable anticyclonic vortices with cyclonic potential vorticity over a seamount, and a corresponding family of cyclonic vortices with anticyclonic potential vorticity over a localized depression. The seamount is described by a characteristic function (corresponding to a flat top) with arbitrary shape.     

A regression study of the number of efficient extreme points in multiple objective linear programming

In this paper we employ regression analysis to construct relationships for predicting the number of efficient extreme points in MOLPs (multiple objective linear programs) with up to 120,000 efficient extreme points, and the CPU time to compute them. Principal among the factors affecting the number of efficient extreme points and CPU time are the number of objectives, criterion cone size, number of constraints, number of variables, and the nonzero density of the constraint matrix. The regression equations show the degree to which interactions are present among the factors and provide a more formal basis for understanding how the complexity of the efficient set, an indicator of the difficulty involved in solving a multiple criteria problem, increases with problem size.     

Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

Structural manipulation of pyrochlores: Thermal evolution of metastable Gd2(Ti1−yZry)2O7 powders prepared by mechanical milling

The structural and microstructural characteristics of metastable Gd₂(Ti_1−yZr_y)₂O₇ powders prepared by mechanical milling have been studied by a combination of XRD and Raman spectroscopy. Irrespective of their Zr content, as-prepared powder phases present an anion-deficient fluorite-type of structure as opposed to the pyrochlore equilibrium configuration obtained for the same solid solution by other synthetic routes. These fluorites are stable versus thermal activation, at least up to temperatures of 800 °C. For the Ti-rich compositions, thermal treatments at higher temperatures facilitate the rearrangement of the cation and anion substructures and the relaxation of mechanochemically induced defects whereas for compositions with high Zr content, the fluorite crystal structure is retained even at temperatures as high as 1200 °C. Interestingly enough, transient pyrochlores showing a very unusual cation distribution were observed during the thermally induced defect-recovery process.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)